Melinda told Gayle King Thursday on "CBS Mornings" ... Bill's relationship with Epstein was definitely a factor in their divorce. You gotta hear the way she describes that relationship -- the disgust in her voice is loud and clear.

Melinda said, "I did not like that he had meetings with Jeffrey Epstein, no. I made that clear to him."

She also tells King she met him a grand total of one time, and it creeped her out ... "I wanted to see who this man was and I regretted it from the second I stepped in the door ... he was evil personified. I had nightmares about it afterward."

She says her heart breaks for the young women who suffered at the hands of Epstein ...  and she says she made her feelings very clear to Bill.

Melinda makes it clear there are still hard feelings between her and Bill ... she says they have a working relationship but are not "friends."

TMZ broke the story ... when they initially planned on announcing the divorce in March 2021, Melinda and their kids went to a private island, anticipating a media onslaught -- and Bill was not invited. The family was furious at him ... we're told largely because of his alleged relationships with other women.